We use the Fermi-LAT gamma-ray observatory to search for gamma-ray emission from four nearby, debris disk-hosting main sequence stars: $tau$ Ceti, $epsilon$ Eridani, Fomalhaut, and Vega. For three stars ($tau$ Ceti, Fomalhaut, and Vega), we establish upper limits that are consistent with theoretical expectations. For $epsilon$ Eridani, we find a possible spatially coincident source with a soft energy spectrum of $dN/dE sim E^{-3.6}$. However, at this stage we are unable to rule out that this emission is due to a more extended feature in the diffuse background. In the interpretation that the emission is due to $epsilon$ Eridani, the $> 100$ MeV gamma-ray luminosity is $sim 10^{27}$ erg/s $simeq 3times 10^{-7}$ L$_odot$, which is $sim 10^{10}$ times the gamma-ray luminosity from the disk of the quiet Sun. We find $lesssim 2 sigma$ evidence of source variability over a $sim 7$ year timescale. In the interpretation that the gamma-ray emission from $epsilon$ Eridani itself, we consider two possible models: 1) cosmic-ray collisions with solid bodies in the debris disk which extends out $sim$60 AU from the host star, and 2) emission from the stellar activity. For the former model, assuming a total disk mass consistent with infrared measurements, we find that the size distribution of bodies is steeper than expected for a collisional cascade state. If confirmed as being associated with $epsilon$ Eridani, this would be the first indication of gamma-ray emission from the vicinity of a main sequence star other than the Sun.
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